Such a reflector lamp is known from WO2008/072131, in which the support body provides accurate positioning of the lamp vessel in the reflector. In the known reflector lamp there is no direct mechanical connection between the lamp vessel and the reflector in the critical area at or adjacent the lamp opening of the reflector, thus causing the reflector lamp to be subjected to lower (thermal) stresses. As the area at and/or adjacent the lamp opening is covered by the support body to only a relatively small extent, free or forced convection at the lamp opening, for example an air flow, is made possible. Thus, the use of high-power lamps is enabled. Lamps of higher power are, for example, short-arc high-pressure discharge lamps having a nominal power of, for example, 250 to 500 W during stable operation, as well as, for example, a UHP lamp designed for a power of 450 W during continuous steady-state operation. It is a disadvantage of the known reflector lamp that relatively high temporary stresses still occur in the reflector, especially during ignition and cooldown of the lamp and in particular when relatively cheap hard glass is used as the material for the reflector. Another disadvantage of the known reflector lamp is that cooling of the lamp is yet relatively inefficient.